That would really annoy me and I would consider it a tad manipulative (but I obviously don't know the person). Having a dog is expensive, time consuming and problematic if you rent and yet your daughter is falling in love. I also think dad should either keep the dog there OR have offered to pay for vets fees and food etc. He's essentially passing on an expense item and yet not offered to support it!
You sound sweet in the sense of being patient and hoping your landlord may accept it.
I know I can be tough on such things BUT to me the father should have broached with you or at least spoken to your landlord first rather than now setting you up with an upset child if the landlord says no etc.
As you suggest you work long shift hours and this would mean the dog would need to be walked otherwise you could come home and find many things ripped and eaten as the dog is bored (plus the potential mess).
Sounds to me like you have enough on your plate and that you need to be firm and wise and assertive and explain the practical issues to your daughter and tell her father he must keep the dog himself. In this way, if he refuses, HE is letting your daughter down and not you. He set this up so he must follow it through (I believe).